Numerical Mathematics of Quasicrystals

Numerical Mathematics of Quasicrystals

Pingwen Zhang

Peking University

Joint work with Kai Jiang

(Xiangtan University)

Outline

Background

Quasicrystals

Mathematics

Experiment

Physics

Numerical Methods

Discussion

Tilings

Tiling: covering the plane without gaps or overlaps

Tilings

Equilateral polygons: Periodic tilings


Tilings

Equilateral polygons: Periodic tilings

Pentagon


gaps

Aperiodic tiling: no invariance by translation

Penrose Tiling

R. Penrose (1974)


Penrose Tiling

A negative example to the Hilbert’s 18th problem

R. Penrose (1974)


Penrose Tiling

A negative example to the Hilbert’s 18th problem

R. Penrose (1974)

Building up of space from congruent polyhedra

The Nobel Prize in Chemistry 2011 was awarded to

Shechtman "for the discovery of quasicrystals".

Quasicrystals

D. Shechtman, I. Blech,

D. Gratias, and J. W. Cahn,

PRL, 53, 1951 (1984).

Complete Order:

Periodic Crystals

Quasicrystals

The role of quasicrystals

Complete Disorder:

Non-Crystalline

Complete Order:

Periodic Crystals

What’s More ?

Quasicrystals


Complete Disorder:

Non-Crystalline

Complete Order:

Periodic Crystals

Quasicrystals

What’s More ?

Quasicrystals


Complete Disorder:

Non-Crystalline

Complete Order:

Periodic Crystals

Quasicrystals Glass

What’s More ?

Quasicrystals


Complete Disorder:

Non-Crystalline

Complete Order:

Periodic Crystals

Quasicrystals Glass

What’s More ?

Quasicrystals


Complete Disorder:

Non-Crystalline

unknown

Mathematics

Physics

Experiment

Numerical Methods

Quasicrystals

Outline

Almost Periodic Function

Periodicity


Periodicity


Periodicity


Periodicity


+

Periodicity


+

Periodicity


+

Periodicity Quasi-Periodicity


+

Periodicity Quasi-Periodicity

In general, Almost Periodic Function

Yves Meyer

Meyer’s work:

1964-1973 QCs from Harmonic Analysis and Number Theory

1974-1984 Calderón-Zygmund operator theory

1983-1993 Wavelet

1994-1999 Navier-Stokes equation

2000-now QCs and Applications

2010 Gauss Prize

2017 Able Prize

Starting point: the local controls the global

Meyer’s work


Meyer’s work

Problems :

How to construct

The properties of


Meyer’s work

Problems :

How to construct

The properties of

A trivial case: is a periodic function

Delone Set

Delone Set

Discrete

Delone Set

Discrete

Relatively dense

Meyer’s QCs

Meyer’s QCs

Meyer’s QCs and Algebraic numbers

Meyer’s QCs

Meyer’s QCs and Algebraic numbers

Meyer’s QCs and Harmonic analysis

Cut-and-Project scheme

A constructive method (Meyer, 1972)

High-D: Periodic Lattice

Window: irrational slice

Low-D: Meyer’s QCs

Cut-and-project scheme can generate tilings.

Dynamical System

Quasi-periodic SchrÖdinger operator

Dynamical System


The dynamical system

Dynamical System



Equivalently

Dynamical System



Equivalently

Sp

ectrum

Dendrimer Liquid QC

The First Soft QCs

X. Zeng, G. Ungar, Y. Liu, V. Percec, A. E. Dulcey, J. K. Hobbs, Nature, 428, 157 (2004).

C12H25

QCs: Discovery

Type of

quasicrystal

d-D n-D Metric Symmetry System First

Report

Icosahedral 3 D 6 5 m 3

_

5_

AlMn Shechtman et al.

1984

Tetrahedral 3D 6 3 m 3

_

AlLiCu Donnadieu, 1994

Decagonal 2D 5 (5) 10/mmm AlMn Chattopadhyay et al.,

1985a and Bendersky,

1985

Dodecagonal 2D 5 3 12/mmm NiCr Ishimasa et al.

1985

Octagonal 2D 5 2 8/mmm VNiSi,

CrNiSi

Wang et al.

1987

Pentagonal 2D 5 (5) 5m_

AlCuFe Bancel , 1993

Natural

Icosahedral

3D 6 (5) m 3

_

5_

khatyrkite Bindi, 2009

Metallic QCs Soft QCs

Soft matter

systems

Type of

quasicrystal

First

Report

Dendrimer liquid

crystals

Dodecagonal Zeng et al.,

2004

ABC star

copolymers

Dodecagonal Hayashida et al.

2007

Colloidal

nanoparticale

Dodecagonal Talapin, 2009

Colloidal coplymer

micelles

nanoparticale

18-fold

Fischer, et al.,

2011

ABA'C tetrablock

terpolymers

Dodecagonal Zhang, Bates,

2012

Mesoporous silica

micelles

Dodecagonal Xiao et al.

2012

Mesoporous silica

nanparticles

Dodecagonal Sun et al. 2017

http://home.iitk.ac.in/~anandh/E-book/Quasicrystals.ppt

2 31

Successive spots are at a distance inflated by

Mg23Zn68Y9 alloy (5-fold)

Experimental QCs

4

2 31

Successive spots are at a distance inflated by

Mg23Zn68Y9 alloy (5-fold)

Experimental QCs

Experimental QCs

∩Meyer’s QCs

4

Quasicrystals : Quasiperiodic Crystals

(Levine-Steinhardt, 1986; Mermin, 1999)

Physical QCs

Diffraction of pattern of

10-fold symmetry in experiment

bj are reciprocal primitive vectors.

b1

b2

b3b4

Numerical Mathematics

Projection method

Crystalline approximant method

One/multi mode approximation

Outline

Finite domains, suitable boundary conditions

Evaluate the energy density to high precision

Put different kinds of ordered structures on an equal footing

Infinite systems without decay

Difficulties in Computing QCs

Diophantine Approximation

Approximate AP function



Diophantine inequality




Continued fraction approximation




Continued fraction approximation

Idea: use a (large) periodic crystal to approximate a QC

Crystalline approximant method (CAM)

Frequency domain: satisfies Diophantine inequality

irrational number

Error : 1. Finity approx. Infinity (Diophantine Approximation error);

2. Numerical error

Error of CAM


2. Numerical error (small)

Error of CAM

2D 12 fold symmetric QC

Computational domain:



Error of CAM


Computational domain:




Error of CAM

ESDA 0.1669 0.0918 0.0374 0.0299 …

L 126 204 3372 53654 …

CAM

CAM: Fourier spectral method

CAM


CAM


CAM


Contradiction

Projection Method: Motivation

Avoid Diophantine Approximation


Avoid Diophantine Approximation



12-fold symmetry in Fourier domain



irrational number



irrational number













Projection matrix:

A d-D QC can be embedded into an n-D periodic structure

Implement in Fourier space



Projection matrix:

Projection Matrix

(1, 0, 0, 0)

(0, 1, 0, 0)

(0, 0, 1, 0)

(0, 0, 0, 1)

Frequency set : high-dimension representation

Projection matrix: connects high-D and low-D

Projection Matrices

2D 8-fold

2D 10-fold

3D Icosahedron

Projection method

Projection Method

Projection method

Assume that

Projection Method

Projection method

Implement in an n-D unit cell

Assume that

Projection Method

Projection method

Periodic crystals: P = Id

Implement in an n-D unit cell

Assume that

Projection Method

In projection method, the energy density can be evaluated by

Theorem: For a d-dimensional quasicrystal , using the

projection method expansion, we have

Projection Method

When study QCs,

Optimize Unit Cell

Presuppose some properties of structures

6-fold symmetry and

One(Multi)-Modes Approximation

Presuppose some properties of structures

6-fold symmetry and

Qualitative analysis

Be available to limited cases

One(Multi)-Modes Approximation

Phase Field Crystal Models

Expansion of the free energy with respect to

Periodic crystals: one length scale

QCs: two length scales

Comparison

Take Lifshitz-Petrich model (PRL, 1997 ) as an example

Projection method can obtain exact energy density.

Comparison



Comparison



Comparison



Projection Method

Comparison



Comparison



No Diophantine Approximation

Projection method (PM) costs less CPU time.

Comparison


Comparison

Modes PM CAM (L=30)

4 -3.469032257481E-03 -3.4712487912555E-03

6 -3.527308744797E-03 -3.4832422187171E-03

8 -3.524067379524E-03 -3.4578379603209E-03

16 -3.524067379522E-03 -3.4578375604919E-03

24 -3.524067379522E-03 -3.4578375604919E-03

32 -3.524067379522E-03 -3.4578375604919E-03

40 -3.524067379522E-03 -3.4578375604919E-03


Comparison

Modes PM CAM (L=30)

4 -3.469032257481E-03 -3.4712487912555E-03

6 -3.527308744797E-03 -3.4832422187171E-03

8 -3.524067379524E-03 -3.4578379603209E-03

16 -3.524067379522E-03 -3.4578375604919E-03

24 -3.524067379522E-03 -3.4578375604919E-03

32 -3.524067379522E-03 -3.4578375604919E-03

40 -3.524067379522E-03 -3.4578375604919E-03

The precision is enough.

Figure: Real space densities and their spectra of (a) 12-fold, (b) 10-fold,

(c) 8-fold QCs computed by the Projection Method in the 4-D space .

QCs in LP Model

Projection method can discover more QCs

Original phase diagram in

Lifshitz-Petrich’s paper

(1997, PRL).

Phase diagram

Stability of 2D QCs

Gaussian-polynomial potential

3D Icosahedral QCs

Diffraction pattern of 3d icosahedral

QCs obtained by projection method

in the 6-D space .

3D Icosahedral QCs



in the 6-D space .

3D Icosahedral QCs

Cu-Ga-Mg-Sc alloys ,

Phil. Mag. Lett. 2002, 82, 483

Compare with experiment



in the 6-D space .

3D Icosahedral QCs

Cu-Ga-Mg-Sc alloys ,

Phil. Mag. Lett. 2002, 82, 483

Compare with experiment

Two-modes approximation method

Phase Diagrams

Two-modes approximation method Projection method

Phase Diagrams


Phase Diagrams

Big

Difference !


Algorithm plays an important role in studying QCs!

Phase Diagrams

Big

Difference !

Discussion and Open Problems

Outline

Riemann Conjecture

Critical line ：

QCs and Riemann Conjecture

Metallic QCs and Soft QCs

Phase transition between crystals/QCs and QCs

…

Problems

What’s a QC in mathematical language ?

Projection method, Cut-and-Projection scheme

Mathematical, Physical, Experimental QCs

Which physical model can produce QCs ?

More structures between perfect crystals and disordered

…

Problems

Thank you !

E-mail: [email protected]

http://www.math.pku.edu.cn/pzhang

Documents

Numerical Mathematics of Quasicrystals